On measures which generate the scalar product in a space of rational functions

Abstract

Let $z_1,z_2,\,\ldots\,,z_n$ be pairwise different points of the unit disc and $\mathscr{L}(z_1,z_2,\,\ldots\,z_n)$ be the linear space generated by the rational fractions $\frac{1}{t-z_1} , \frac{1}{t-z_2} , \cdots\ , \frac{1}{t-z_n}\cdot$ Every non-negative measure $\sigma$ on the unit circle $\mathbb{T}$ generates the scalar product \[\langle\,f\,,\,g\,\rangle_{\!_{L^2_\sigma}} =\int\limits_{\mathbb{T}}f(t)\,\bar{g(t)}\,\sigma(dt), \quad \forall\,f,g\,\in\,L^2_\sigma.\] The measures $\sigma$ are described which satisfy the condition \[\langle\,f\,,\,g\,\rangle_{\!_{L^2_\sigma}}= \langle\,f\,,\,g\,\rangle_{\!_{L^2_m}},\quad \forall\,f,g\in\mathscr{L}(z_1,z_2,\,\ldots\,z_n),\] where $m$ is the normalized Lebesgue measure on $\mathbb{T}$.